The following system of equations is represented by the matrix equation $\text{A}\left[\begin{array} {ccc} x \\ y \\ z \end{array} \right]=\vec{b}$. $\begin{aligned}20x+35y-10z&=15 \\16x+8y-4z&=12 \\3x-6y+9z&=18\end{aligned}$ ${A}=$ $\vec{b}=$ Represent each row and column in the order in which the variables and equations appear.
The Strategy A system of equations can be represented by a matrix equation $\text{A}\vec{x}=\vec{b}$, where $\text{A}$ is the coefficient matrix, $\vec{x}$ is the variables vector, and $\vec{b}$ is the constants vector. Each row of the matrix equation represents an equation in the system. [I need an explanation, please!] Representing the system of equations as a matrix equation We are given the system of equations: $\begin{aligned}20x+35y-10z&=15 \\16x+8y-4z&=12 \\3x-6y+9z&=18\end{aligned}$ First, let's rewrite this system to show the coefficients of each variable. $\begin{aligned}{20}x+{35}y+({-10})z&=15 \\{16}x+{8}y+({-4})z&=12 \\{3}x+({-6})y+{9}z&=18\end{aligned}$ Now, the coefficient matrix can be written as follows. $\left[\begin{array} {ccc} {20} & {35} & {-10} \\ {16} & {8} & {-4} \\ {3} & {-6} & {9} \end{array} \right]$ We can multiply this matrix by a column vector of variables and set it equal to a column vector with the values on the right side of the equations, as follows. $\left[\begin{array} {ccc} {20} & {35} & {-10} \\ {16} & {8} & {-4} \\ {3} & {-6} & {9} \end{array} \right]\left[\begin{array} {ccc} x \\ y \\ z \end{array} \right] =\left[\begin{array} {ccc} 15 \\ 12 \\ 18 \end{array} \right]$ This is our matrix equation. Summary $\text{A}$ and $\vec{b}$ are shown below. $\text{A}=\left[\begin{array} {ccc} 20 & 35 & -10 \\ 16 &8 & -4 \\ 3 & -6 & 9 \end{array} \right]~~~~~~~~~~~~ \vec{b}=\left[\begin{array} {ccc} 15 \\ 12 \\ 18 \end{array} \right]$